Adiabatic Limits of Co-Associative Kovalev-Lefschetz Fibrations

Adiabatic limits of co-associative Kovalev-Lefschetz ﬁbrations

Simon Donaldson April 22, 2016

To Maxim Kontsevich, on his 50th. birthday

1 Introduction

This article makes the first steps in what we hope will be a longer project, inves- tigating seven-dimensional G2-manifolds from the point of view of co-associative fibrations, and in particular the “adiabatic limit”, when the diameters of the fibres shrink to zero. To set up the background, recall that there is a notion of a “positive” exterior 3-form on an oriented 7-dimensional real vector space E, and that such a form defines a Euclidean structure on E (definitions are given in Section 2 below). Thus there is a notion of a positive 3-form φ on an oriented 7-manifold M, which defines a Riemannian metric gφ and hence a 4-form ∗φφ (writing ∗φ for the Hodge ∗-operator of gφ). A torsion-free G2-structure can be defined to be a positive form φ such that φ and ∗φφ are both closed. (We refer to [14] for the terminology and background.) We can also consider the weaker condition of a closed G2-structure in which we just require that the 3-form φ is closed. We have then various fundamental questions, such as the following. 1. Given a compact oriented 7-manifold and a de Rham cohomology class C ⊂ Ω3(M), does C contain a positive form? In other words, is there a non-empty subset C+ ⊂ C of positive forms?

+ 2. If so, is there a torsion free G2-structure in C , i.e. can we solve the + equation d ∗φ φ = 0 for φ ∈ C ? 3. The Hitchin functional H on the space of positive 3-forms is just the volume H(φ) = Vol(M, gφ).

The equation d ∗φ φ = 0 is the Euler-Lagrange equation for this functional restricted to C+—in fact critical points are local maxima [11]. It is then

1 interesting to ask if H is bounded above on C+ and, if a critical point exists, whether it is a global maximum. 4. Bryant [4] deﬁned a “Laplacian” ﬂow on C+: ∂φ = d (∗ d ∗ φ) , ∂t φ φ

whose fixed points are torsion-free G2 structures. It is interesting to ask if this flow converges as t → ∞ to a fixed point. While these are obvious and natural questions, any kind of systematic answers seem far out of reach at present. In this article we will study 7-manifolds with extra structure given by a “Kovalev-Lefschetz fibration” π : M → S3 with co-associative fibres. These fibres will be K3 surfaces, either smooth or with nodal singularities. The idea underlying our discussion is that there should be an “adiabatic limit” of the torsion-free G2-condition, and more generally of each of the questions above. Leaving for the future any precise statement along those lines we will write down precise definitions which we propose as limiting geometric objects and we will see that questions (1)-(4) have sensible analogs in this setting. Co-associative fibrations were studied by Gukov, Yau and Zaslow [9] and there is overlap between our discussion in Section 2 and theirs. More recent work by Baraglia [2] in the case of torus fibrations is especially relevant to our constructions in this paper (see also the brief comments in 5.1 below) In particular Baraglia made the connection with maximal submanifolds in spaces with indefinite signature, which is the fundamental idea that we use here. The notion of a Kovalev-Lefschetz fibration was essentially introduced by Kovalev in [16]. In other directions there is a substantial literature on adiabatic limits of various kinds of structures in Riemannian geometry. Examples include constant scalar curvature Kählermetrics [5] and many papers by different authors on “collapsing” Calabi-Yau manifolds, such as [8], [18]. We will not attempt to give a survey of existence results for compact G2-manifolds here, but we mention the landmark results of Joyce [13], Kovalev [15] and Corti, Haskins, Nordström, Pacini [6].

2 Co-associative ﬁbrations

We review some standard multi-linear algebra background. Let E be an oriented 7-dimensional real vector space and let φ be a 3-form in Λ3E∗. We have then a quadratic form on E with values in the oriented line Λ7E∗ deﬁned by

2 Gφ(v) = (v * φ) ∧ φ. (1)

We say that φ is a positive 3-form if Gφ is positive deﬁnite. In this case, any choice of (oriented) volume form makes Gφ into a Euclidean structure on E and there is a unique choice of the volume form such that |φ|2 = 7. So we now have a Euclidean structure Gφ canonically determined by a positive 3-form φ.

2 For (E, φ) as above we say that a 4-dimensional subspace V ⊂ E is co- associative if the restriction of φ to V vanishes. A standard model for such a triple (E, φ, V ) can be built as follows. We start with R4 with co-ordinates 2 x0, . . . , x3 and standard metric and orientation. Let Λ+ be the space of self-dual 2-forms. This has a standard basis ω1, ω2, ω3 given by

ωi = dx0dxi + dxjdxk where i, j, k run over cyclic permutations of 1, 2, 3. Let t1, t2, t3 be the co- 2 ordinates on Λ+ corresponding to this basis and set X φ0 = ωidti − dt1dt2dt3, (2) i

3 ∗ 4 2 ∗ an element of Λ (E0) where E0 = R ⊕ (Λ+) . This is a positive 3-form if E0 is given the orientation

−dt1dt2dt3dx0dx1dx2dx3dx4 (3) and the corresponding Euclidean structure is the standard one. The subspace 4 R ⊂ E0 is co-associative and any co-associative triple (E, φ, V ) is equivalent 4 to the model (E0, φ0, R ) by an oriented linear isomorphism from E to E0. Somewhat more generally, let V be an oriented 4-dimensional real vector space. The wedge product gives a quadratic form on 2-forms up to a positive factor. Let H be a 3-dimensional vector space and ω ∈ H∗ ⊗ Λ2V ∗. We say that ω is hypersymplectic if, regarded as a linear map from H to Λ2V ∗, it is an injection to a maximal positive subspace for the wedge product form. There is an induced orientation of H and, with a suitable sign convention, for any 3 ∗ oriented volume form volH ∈ Λ H the 3-form

φ = ω + volH (4) in H∗ ⊗Λ2V ∗ ⊕Λ3H∗ ⊂ Λ3(V ⊕H)∗ is positive. Moreover the general positive 3- form on V ⊕H such that V is co-associative and H is the orthogonal complement of V with respect to the induced Euclidean structure has the shape above. Notice that a hypersymplectic ω deﬁnes a conformal structure on V —the unique structure such that the image of H is the self-dual subspace. Next we consider an oriented 7-manifold M. As stated in the introduction, a closed G2-structure on M is a closed 3-form φ which is positive at each point and the structure is torsion-free if in addition d ∗φ φ = 0. If φ is any positive 3-form a co-associative submanifold is a 4-dimensional submanifold X ⊂ M such that each point p ∈ X the tangent space TXp is a co-associative subspace of TMp with respect to φ(p) [10]. The form φ induces a standard orientation of X, for example by declaring that ∗φφ is positive on X. As background, we recall that for general positive forms the co-associative condition is over-determined, but for closed forms φ there is an elliptic deformation theory and compact co- associative submanifolds are stable under small perturbations of the 3-form [17]. They are parametrised by a moduli space of dimension b+(X) (the dimension

3 of a maximal positive subspace for the intersection form). If φ is torsion-free then co-associative submanifolds are examples of Harvey and Lawson’s calibrated submanifolds[10], with calibrating form ∗φφ. We pause here to note a useful identity. Lemma 1 If φ is a closed positive form then for any vector v in TM we have

(v * φ) ∧ d ∗φ φ = (v * dφ) ∧ ∗φφ. This can be derived from the diﬀeomorphism invariance of the Hitchin functional H(φ) = Vol(M, gφ). The ﬁrst variation under a compactly supported variation is 1 Z δH = (δφ) ∧ ∗φφ. 3 M

If v is a compactly supported vector on M and δφ is the Lie derivative Lvφ then clearly the ﬁrst variation is zero. Writing Lvφ = d(v * φ) + v * dφ and integrating by parts we get the identity Z (v * dφ) ∧ ∗φφ − (v * φ) ∧ d ∗φ φ = 0, M which implies the result.

We now move on to our central topic—fibrations π : M → B of a 7-manifold over a 3-dimensional base with co-associative fibres. Later we will consider “fibrations” in the algebraic geometers sense, with singular fibres, but in this section we will consider a genuine locally trivial bundle, with fibre a 4-manifold X. For definiteness we take X to be the smooth 4-manifold underlying a K3 surface, with the standard orientation. We start with a fixed C∞ fibration π : M → B where M and B are oriented. We want to work with differential forms on the total space and to this end we recall some background. Let V ⊂ TM be the tangent bundle along the fibres, so we have an exact sequence 0 → V → TM → π∗(TB) → 0. For p, q ≥ 0 let Λp,q be the vector bundle

Λp(π∗T ∗B) ⊗ ΛqV ∗ over M and write Ωp,q for the space of sections of Λp,q. The exterior derivative on the ﬁbres is a natural diﬀerential operator, for p, q ≥ 0:

p,q p,q+1 df :Ω → Ω ,

∗ ∗,∗ and there is a canonical ﬁltration of ΩM with quotients the Ω . Now suppose that we have a connection H on π : M → B. That is, a subbundle H ⊂ TM with TM = V ⊕ H. Then we get a direct sum decomposition

∗ M p,q ΩM = Ω . p,q≥0

4 ∗ ∗ The exterior derivative d :ΩM → ΩM has three components with respect to this decomposition.

1. The exterior derivative along the ﬁbres df , as above. 2. The coupled exterior derivative in the horizontal direction

p,q p+1,q dH :Ω → Ω . This can be defined in various ways. One way is to work at a given point b ∈ B and to choose a local trivialisation of the fibration compatible −1 with H on π (b). Then, over this fibre, dH is given by the horizontal component of the exterior derivative in this trivialisation. From another point of view we can regard H as a connection on a principal bundle with structure group Diff(X). Then dH is the usual coupled exterior derivative on an infinite dimensional associated vector bundle.

3. The curvature term. The curvature of the connection H is a section FH of the bundle V ⊗ π∗Λ2T ∗B deﬁned by the equation

[ξ1, ξ2]V = FH (ξ1, ξ2),

for any sections ξ1, ξ2 of H, where [ , ]V denotes the vertical component of the Lie bracket. (This agrees with ordinary notion of curvature if we take the point of view of a Diﬀ(X) bundle.) The tensor FH acts algebraically on Ωp,q by wedge product on the horizontal term and contraction on the vertical term, yielding a map (which we denote by the same symbol)

p,q p+2,q−1 FH :Λ → Λ .

The fundamental diﬀerential-geometric formula for the exterior derivative on the total space is then

p,q p,q+1 p+1,q p+2,q−1 d = df + dH + FH :Ω → Ω + Ω + Ω . (5) We say that ω ∈ Ω1,2 is a hypersymplectic element if it is hypersymplectic at each point, as deﬁned above.

Proposition 1 Given π : M → B as above, a closed G2 structure on M such that the ﬁbres of π are co-associative, with orientation compatible with those given on Mand B, is equivalent to the following data. 1. A connection H.

1,2 2. A hypersymplectic element ω ∈ Ω with df ω = 0, dH ω = 0. 3. A tensor λ ∈ Ω3,0 with df λ = −FH ω (6) and such that the value of λ at each point is positive, regarded as an element of Λ3T ∗B.

5 This proposition follows immediately from the algebraic discussion at the beginning of this section and the description of the exterior derivative above. The corresponding positive 3-form φ is of course ω + λ. Note that if we have any 3-form φ which vanishes on the fibres there is a well-defined component ω of φ in Ω1,2, independent of a connection. This just ∗ reflects the naturality of the filtration of ΩM . If φ is closed then df ω = 0. At this point it is convenient to switch to a local discussion (in the base), so we suppose for now that B is a 3-ball with co-ordinates t1, t2, t3. Suppose we have any closed 3-form φ on M which vanishes on the fibres. Then the de Rham cohomology class of φ is zero and we can write φ = dη for a 2-form η on M. By assumption, the restriction of dη to each fibre is zero so we get a map

h : B → H2(X), (7) taking a point to the cohomology class of η in the ﬁbre over that point. Changing the choice of η changes h by the addition of a constant in H2(X). As discussed above, the form φ has a well-deﬁned component ω ∈ Ω1,2 which we can write

3 X ω = ωidti, i=1 where the ωi are closed 2-forms on the fibres. If we fix a trivialisation of the bundle then the ωi become families of closed 2-forms on a fixed 4-manifold X, parametrised by (t1, t2, t3). A moments thought shows that the de Rham cohomology class [ωi] is ∂h [ωi] = . (8) ∂ti This formula makes intrinsic sense, since the cohomology groups of the different fibres are canonically identified. We say that a map h : B → H2(X, R) is positive if it is an immersion and the image of its derivative at each point is a positive subspace with respect to the cup product form. Then it is clear that a necessary condition for the existence of a hypersymplectic element ω representing the derivative is that h is positive.

We now change point of view and ask how we can build up a form φ. We start with a positive map h : B → H2(X) and suppose that we have chosen a P 1,2 hypersymplectic element ω = ωidti ∈ Ω representing the derivative of h.

Lemma 2 Given ω as above there is a connection H such that dH ω = 0.

We work in a local trivialisation so that the ωi are regarded as t-dependent 2- forms on the ﬁxed 4-manifold X. The connection is represented by t-dependent vector ﬁelds v1, v2, v3 on X and the condition that dH ω = 0 is

∂ωi ∂ωj − + (Lvj ωi − Lvi ωj) = 0 (9) ∂tj ∂ti

6 ∂h By assumption, ωi represents the cohomology class and this implies that the ∂ti ∂ω cohomology class of ∂ωi − j is zero. So we can choose t-dependent 1-forms ∂tj ∂ti a1, a2, a3 such that ∂ωi ∂ωj − = dak, ∂tj ∂ti for i, j, k cyclic. Since the ωi are closed we can write (9) as

d(vj * ωi − vi * ωj) = dak, which is certainly satisﬁed if

vj * ωi − vi * ωj = ak (10)

Let S : V ⊕ V ⊕ V → V ∗ ⊕ V ∗ ⊕ V ∗ be the map which takes a triple of tangent vectors v1, v2, v3 to a triple of cotangent vectors vj * ωi − vi * ωj. What we need is the elementary linear algebra statement that S is an isomorphism. To see this observe that it is a pointwise statement and by making a linear change of co-ordinates ti we can 4 reduce to the case when ωi are the standard basis of the self-dual forms on R . Using the metric to identify tangent vectors and cotangent vectors, S becomes the map which takes a triple v1, v2, v3 to the triple

Jv3 − Kv2 , Kv1 − Iv3 , Iv2 − Jv1. (11)

Here we make the standard identiﬁcation of R4 with the quaternions and use quaternion multiplication. A short calculation in quaternion linear algebra shows that S is an isomorphism. Once we have chosen ω and H, as above, the remaining data we need to 3,0 construct a closed G2 structure with co-associative ﬁbres on M is λ ∈ Ω which we can write in co-ordinates as −λdt1dt2dt3 for a function λ on M. The 2 ∗ identity d = 0 on ΩM implies that

df (FH ω) = 0.

Since H1(X) = 0 we can solve the equation

df λ = −FH ω.

The solution is unique up to the addition of a lift of a 3-form from B, i.e. up to changing λ by the addition of a function of t. By adding a suﬃciently positive function of t we can arrange that λ is positive. We conclude that any choice of hypersymplectic element representing the derivative of h can be extended to a closed G2 structure with co-associative ﬁbres.

We stay with the local discussion and go on to consider torsion-free G2 structures. Suppose we have 2-forms ωi spanning a maximal positive subspace

7 of the vertical tangent space V . Let χ be an arbitrary volume form on V , so that (ωi ∧ ωj)/χ are real numbers. We form the determinant det((ωi ∧ ωj)/χ) and the 4-form 1/3 (det((ωi ∧ ωj)/χ))) χ. It is clear that this is independent of the choice of χ and we just write it as 1/3 det (ωi ∧ ωj). Lemma 3 If φ = ω + λ ∈ Ω1,2 + Ω3,0 is a positive 3-form on M, as considered above, then ∗φφ = Θ + µ where

2,2 P 1. Θ = F1(ω, λ) ∈ Ω is given by cyclic Θidtjdtk where Θi are forms in the linear span of ω1, ω2, ω3 determined uniquely by the condition that

1/3 1/3 Θi ∧ ωj = δij(λ det (ωa ∧ ωb)); (12)

4,0 2. µ = F2(ω, λ) ∈ Ω is given by

1/3 −2/3 det (ωa ∧ ωb)λ . (13)

This is a straightforward algebraic exercise using the deﬁnitions. The upshot of our local discussion is then the following statement.

Proposition 2 A torsion-free G2 structure on M with co-associative fibres is given by a map h (defined up to a constant), a hypersymplectic element ω representing the derivative of h, a connection H and λ ∈ Ω3,0 with λ > 0, satisfying the five equations

1. dH ω = 0,

2. df λ = −FH ω,

3. dH µ = 0,

4. df Θ = −FH µ,

5. dH Θ = 0. Here µ, Θ are determined algebraically from ω, λ by the formulae (12),(13).

Note that the equation (3) in Proposition 2 has a clear geometric meaning. It states that the structure group of the connection reduces to the volume preserving diﬀeomorphisms of X. It is also equivalent to the statement that the ﬁbres are minimal submanifolds, with respect to the metric gφ. We will not attempt to analyse these equations further in this form, except to note one identity.

8 Lemma 4 Suppose that h, ω, λ,H satisfy equations (1), (2), (3), (4) in Propo- sition 2. Then dH Θ = Gdt1dt2dt3 where G is an anti-self dual harmonic form on each ﬁbre with respect to the conformal structure determined by ω.

2 First, the identity d ∗φφ = 0 implies that, under the stated conditions df (dH Θ) = 0 so dH Θ = Gdt1dt2dt3 where G is a closed 2-form on each ﬁbre. Now we apply Lemma 1, taking v to be a horizontal vector. Since dφ = 0 we have 0,2 2,0 (v * φ) ∧ d ∗φ φ = 0 . The contraction v * φ has terms in Ω and Ω but only the ﬁrst contributes to the wedge product with d∗φ φ and the result follows immediately.

Let us now move back to consider the “global” case, with a K3 fibration π : M → B over a general oriented 3-manifold B. We have a flat bundle H → B with fibre H2(X; R) induced from an H2(X; Z) local system, so there is an intrinsic integer lattice in each fibre. Let φ be a closed 3-form on M, vanishing on the fibres. If we cover B by co-ordinate balls Bα we can analyse −1 φ on each π (Bα) as above, starting with a choice of a smooth section hα of

H|Bα . Then on Bα ∩ Bβ the diﬀerence

χαβ = hα − hβ (14) is a constant section of H. This deﬁnes a Cech cocycle, and hence a cohomology class χ ∈ H1(B; H), where H is the sheaf of locally constant sections of the ﬂat bundle H. This class χ is determined by the de Rham cohomology class of φ. In fact the Leray-Serre spectral sequence gives us an exact sequence

0 → H3(B) → H3(M) →b H1(B; H) → 0, (15) and [φ] ∈ H3(M) maps to χ ∈ H1(B; H). It is convenient to express this in terms of an “aﬃne” variant of H. Let G be the group of aﬃne isometries of H2(X), an extension

1 → H2(X) → G → O(H2(X)) → 1.

1 2 A class χ ∈ H (B; H) defines a flat bundle Hχ with fibre H (X) and structure group G over B. One way of defining this is to fix a cover and Cech representative χαβ and then decree that locally constant sections of Hχ are given by constant sections hα over Bα satisfying (14). In these terms the conclusion is that if we start with a class Φ ∈ H3(M), map to a class χ = b(Φ) and form 3 the bundle Hχ, then any representative φ ∈ Ω (M) for Φ which vanishes on the fibres defines a canonical smooth section h of the flat bundle Hχ. Moreover if φ is a positive 3-form then h is a positive section of Hξ, using the obvious extension of the definition in the local situation.

9 2.1 The adiabatic limit We have seen that the torsion-free condition for a positive 3-form involves six equations for a triple (ω, λ,H)

df ω = 0, dH ω = 0, df λ = −FH ω, dH µ = 0, df Θ = −FH µ, dH Θ = 0. where Θ = F1(ω, λ) and µ = F2(ω, λ) are determined algebraically by ω, λ. We introduce a positive real parameter and write ω = ω˜,

2 Θ˜ = F1(˜ω, λ) = Θ, µ˜ = F2(˜ω, λ) = µ. Then the six equations, written for (˜ω, λ,H) become

d ω˜ = 0, d ω˜ = 0, d λ + F ω˜ = 0, f H f H (16) dH µ˜ = 0, df Θ˜ + FH µ˜ = 0, dH Θ˜ = 0.

Hereµ ˜, Θ˜ are related toω ˜, λ in just the same way as µ, Θ were to ω, λ. Geometrically, in the metric determined byω ˜, λ the volume of the ﬁbre is 2 times that in the metric determined by ω, λ. For non-zero these equations are completely equivalent to the original set; the fundamental idea we want to pursue is to pass to the “adiabatic limit”, setting = 0 so that the terms involving the curvature FH drop out. At this stage we simplify notation by dropping the tildes, writing ω, Θ, µ forω ˜, Θ˜ , µ˜.

We say that ω ∈ Ω1,2 is a hyperkählerelement if, in local co-ordinates on the P base, ω = ωidti with ωi ∧ ωj = aijν, where ν is a volume form along the fibres and aij is a positive definite matrix constant along the fibres. Lemma 5 Suppose that a pair (ω, λ) satisfies the five equations

df ω = 0, dH ω = 0, df λ = 0, dH µ = 0, df Θ = 0. Then ω is a hyperkähler element and λ is the lift of a positive 3-form on B. Conversely given a hyperkählerelement ω and a positive 3-form on B there is a unique connection H which yields a solution to these equations. In one direction, suppose that we have ω, λ satisfying these equations and P P work locally, writing ω = ωidti, λ = −λdt1dt2dt3, Θ = Θidtjdtk. The equation df λ = 0 immediately gives that λ is a lift from B. We know that + Θi takes values in the self-dual space Λ spanned by ω1, ω2, ω3 so the equation + df Θ = 0 says that the Θi are self-dual harmonic forms. Since b (X) = 3 we P have Θi = gijωj where the functions gij are constant on the fibres. Now

1/3 1/3 Θi ∧ ωk = λ δik det (ωa ∧ ωb),

10 so X 1/3 1/3 gijωj ∧ ωk = λ δik det (ωa ∧ ωb) j which implies that

jk 1/3 1/3 ωj ∧ ωk = g λ det (ωa ∧ ωb) where gjk is the inverse matrix, also constant on the fibres. Hence ω is hy- perkähler. For the converse, suppose that we have a hyperkählerelement ω and volume form on B, which we may as well take to be −dt1dt2dt3 in local co-ordinates. 1/3 Then we have a volume form µ along the fibres given by det (ωi ∧ ωj) and we have to show that there is a unique connection H such that dH ω = 0 and dH µ = 0. For this we go back to the proof of Lemma 2. We can choose a local trivialisation such that the volume form along the fibres is constant. For any choice of 1-forms a1, a2, a3 with

∂ωi ∂ωj − = dak ∂tj ∂ti we showed there that there is a unique choice of connection H. But we are free to change the 1-forms ak by the derivatives of functions. It is easy to check that the condition that the connection is volume preserving is exactly that the ∗ 1-forms ak satisfy d ak = 0, with respect to the induced metric on the fibres. By Hodge theory, this gives the existence and uniqueness of the connection H. (In fact the argument shows that for any volume form along the fibres we can choose H to satisfy this additional condition, provided that the volume of the fibres is constant.)

We can clearly choose the solution in Lemma 5 so that the µ-volume of the ﬁbres is 1, and we ﬁx this normalisation from now on.

2 It remains to examine the last equation, dH Θ = 0. Recall that H (X, R) carries an intrinsic quadratic from of signature (3, 19). This defines a volume form on any submanifold and hence an Euler-Lagrange equation defining sta- tionary submanifolds. In our context we are interested in 3-dimensional sta- tionary submanifolds whose tangent spaces are positive subspaces and these are called maximal submanifolds. A section h of Hξ can be represented locally by an embedding of the base in H2(X) and we say that h is a maximal section if the image is a maximal submanifold. More generally, for any section h the mean curvature m(h) is defined as a section of the vector bundle (Im dh)⊥ ⊂ H which is the normal bundle of the submanifold.

Lemma 6 Suppose that h is a positive section of Hχ, ω is a hyperk¨ahlerelement representing the derivative of h, and that H, λ are chosen as above. Then

dH Θ = m(h)λ, (17)

11 where we use the metric on the fibres defined by ω to identify (Im dh)⊥ with the anti-self-dual 2-forms along the fibres and use the result of Lemma 4. In particular, dH Θ = 0 if and only if h is a maximal positive section. R In local co-ordinates ti let gij = X ωi ∧ωj and set J = det(gij). The volume of the image is given by Z 1/2 J dt1dt2dt3 and the mean curvature is

−1/2 1/2 ij m = J J g h,j , ,i

ij where g is the inverse matrix and ,i denote partial diﬀerentiation. That is

X ∂Θ∗ m = J −1/2 i , ∂ti

∗ 1/2 ij where Θi = J g [ωj]. Computing in a trivialisation compatible with H at a given point we have X ∂Θi dH Θ = dt1dt2dt3. ∂ti

Now Θi is deﬁned by the identity

1/3 1/3 Θi ∧ ωj = det λ δij, so Z Z 1/3 1/3 Θi ∧ ωj = λ det δij, X X whereas Z ∗ 1/2 Θi ∧ ωj = J . X The condition that the µ-volume of the ﬁbres is 1 is Z λ2/3 = det1/3. X On the other hand one has Z det1/3 = J 1/3. X

∗ −1/2 This implies that Θi = [Θi] and J dt1dt2dt3 = λ and the result follows.

One can also prove this Lemma more synthetically, comparing the Hitchin 7-dimensional volume functional with the 3-dimensional volume of the image of h.

12 2.2 Formal power series solutions The conclusion that we are lead to in the preceding discussion is that the adiabiatic limit of the torsion-free G2-structure condition should be the maximal submanifold equation m(h) = 0. To give more support to this idea we show that a solution of the latter equation can be a extended to a formal power series solution of system of equations (16). For simplicity we work locally, so we take the base B to be a ball. Proposition 3 1. Suppose that the quadruple (h, ω, H, λ) satisﬁes the ﬁve equations of Lemma 5. Then there are formal power series

∞ ∞ ∞ X X X ω˜ = ω + ω(j)j,H = H + ζ(j)j, λ = λ + λ(j)j j=1 j=1 j=1 which give a formal power series solution of the first five equations of (16). We can choose ω(j) to be exact, so that ω˜ represents the derivative of the fixed map h. 2. Suppose that in addition h satisfies the maximal submanifold equation m(h) = 0 and is smooth up to the boundary of the ball B. Then we can find a formal power series

∞ X h = h + h(j)j j=1 and power series ω˜,H, µ as above but now with ω˜ representing the derivative of h and such that the quadruple (h, ω˜,H, λ) is a formal power series solution of the equations (16). The proof follows standard lines once we understand the linearisation of the equations, and for this we to develop some background. Let ωi be a standard hyperkählertriple on X. Consider an infinitesimal variation ωi +ηi with dηi = 0 + − + P and write ηi = ηi + ηi in self-dual and anti-self-dual parts. So ηi = Sijωj for some co-efficients Sij. A short calculation shows that the variation in df Θ P is, to first order, dTij ∧ ωj where 2 T = S + S − Tr(S)δ . (18) ij ij ji 3 ij (i.e. T is obtained from S by projecting to the symmetric, trace-free component.) We want to vary ωi in fixed cohomology classes, by dai. The resulting 1 3 3 3 first order variation in dΘi is given by a linear map L1 from ΩX ⊗R to ΩX ⊗R . So the calculation above shows that this linear map is a composite

+ 1 3 d ⊗1 2,+ 3 A 2,+ 3 d⊗1 3 3 ΩX ⊗ R → ΩX ⊗ R → ΩX ⊗ R → ΩX ⊗ R , (19) P P where A is the map taking Sijωj to Tijωj, with T given by (18) above. 2,+ 3 Thus the image of A is the subspace of ΩX ⊗R consisting of tensors with values

13 2 3 in s0(R )—symmetric and trace-free. Here of course we are using repeatedly the fact that the bundle of self-dual forms is identified with R3 by the sections ωi. From another point of view, write S+,S− for the spin bundles over X and p p write S+,S− for their pth. symmetric powers. Of course in our situation S+ is 2 3 a trivial flat bundle. The tensors with values in s0(R ) can be identified with 4 4 sections of S+. (More precisely, S+ has a real structure and we should consider real sections, but in the discussion below we do not need to distinguish between 3 3 3 the real and complex cases.) There is a contraction map from ΩX ⊗ R to ΩX given by X P (ψ1, ψ2, ψ3) = (∗ψi) ∧ ωi (20)

3 The kernel of this contraction map can be identiﬁed with S+⊗S−, corresponding to the decomposition:

2 3 (S+ ⊗ S−) ⊗ S+ = (S+ ⊗ S−) ⊕ (S+ ⊗ S−). Starting with 2,+ 3 3 3 d ⊗ 1 : ΩX ⊗ R → ΩX ⊗ R , 4 3 we first restrict to sections of S+ and then project to sections of S+ ⊗ S−. This defines a differential operator

4 3 δ1 : Γ(S+) → Γ(S+ ⊗ S−). On the other hand, starting with

3 3 4 3 d :ΩX ⊗ R → ΩX ⊗ R ,

3 we restrict to sections of S+ ⊗ S− to deﬁne

3 2 δ2 : Γ(S+ ⊗ S−) → Γ(S+). So we have 4 δ1 3 δ2 2 Γ(S+) → Γ(S+ ⊗ S−) → Γ(S+). The key fact we need is that this is exact at the middle term, i.e. that

Ker δ2 = Im δ1. (21)

∗ In fact, calculation shows that δ2 + δ1 can be identiﬁed with the Dirac operator 3 coupled with S+, i.e.

3 3 D : Γ(S+ ⊗ S−) → Γ(S+ ⊗ S+) using the isomorphism 3 + + S+ ⊗ S+ = S4 ⊕ S2 .

Since S+ is trivial this Dirac operator is a sum of copies of the ordinary Dirac operator over the K3 surface and has no kernel in the S− term. This implies

14 (21). We can say a bit more. The kernel of δ1 consists of the constant sections 4 + of S+ so for any ρ ∈ S3 ⊗ S− we can solve the equation δ1T = ρ with T in the L2-orthogonal complement of the constants. But this means that we can solve the equation (d+ ⊗ 1)a = T . To sum up we have shown:

3 3 Lemma 7 If ρ ∈ ΩX ⊗ R satisﬁes the two conditions • P (ρ) = 0 where P is the contraction (20);

4 3 • (d ⊗ 1)(ρ) = 0 in ΩX ⊗ R ; 1 3 3 3 then ρ is in the image of L1 :ΩX ⊗ R → ΩX ⊗ R .

We now turn to the proof of the ﬁrst item in Proposition 3. At stage k we suppose that we have found ω(i), λ(i), ζ(i) for i ≤ k so that the ﬁnite sums

k X ω[k] = ω + ω(k) etc., i=1 satisfy the first five equations of (16) modulo k+1. We want to choose ω(k+1), λ(k+1) and ζ(k+1) so that if we define

ω[k+1] = ω[k] + k+1ω(k+1) etc. we get solutions of the first five equations of (16) modulo k+2. The essential point is that this involves solving linear equations for ω(k+1), λ(k+1), ζ(k+1), and moreover these equations are essentially defined fibrewise.

Step 1 Let E3,1 be the “error term”

[k] [k] k+1 k+2 df λ + FH[k]ω = E3,1 + O( ).

(Here the O( ) notation means in the sense of formal power series expansion.) The equation we need to solve to correct this error term is

(k+1) df λ = −E3,1, involving only λ(k+1). The identity d2 = 0 on M implies that

[k] [k] df FH[k]ω = −dH dH[k] ω

k+1 which is O( ). This means that df E3,1 = 0 so we can solve the equation for λ(k+1). The solution is unique up to the addition of a term which is constant on the ﬁbres.

15 Step 2 Now let E2,3 be the error term

[k] [k] k+1 k+2 df Θ + FH[k]µ = E2,3 + O( ). The equation we need to solve to correct this error involves the first order [k] variation of df Θ with respect to changes in ω and λ. For the first, we write ω(k+1) as the derivative of a ∈ Ω1 ⊗ R3 as above and the first order variation is given by L1(a). Denote the variation with respect to λ by L2. The equation we need to solve is (k+1) L1(a) + L2(λ ) = −E2,3.

The same argument as before shows that df E2,3 = 0. We do not need to write (k+1) our L2 explicitly because it is easy to see that L2(λ ) only involves the derivative along the fibre, so is unchanged if we change λ(k+1) by a term which (k+1) is constant on the fibres. Thus L2(λ ) is fixed by Step 1 and we have to solve the equation L1(a) = ρ, (k+1) for a, where ρ = −E2,3 − L2(λ ). The same argument as before shows that df E2,3 = 0 and it is clear that df ◦ L2 = 0. Thus df ρ = 0. We claim that, on each fibre, P (ρ) = 0. Given this claim, Lemma 7 implies that we can find the desired solution a. To verify the claim we use the identity of Lemma 1. We take v to be a vertical vector and φ to be the 3-form

φ = ω[k] + λ[k+1].

By what we have arranged, the (3, 1) component of dφ is O(k+2) and this means that (v * ω[k]) ∧ ρ is O(). In turn this implies that (v * ω) ∧ ρ vanishes for all vertical vectors v, and it is easy to check that this is equivalent to the condition P (ρ) = 0.

Step 3 At this stage we have chosen ω(k+1). We still have the ambiguity in λ(k+1) up to constants along the fibres. We fix this by decreeing that the fibres have volume 1+O(k+2) with respect to the form µ[k+1] defined by λ[k+1], ω[k+1].

[k+1] [k+1] [k+1] Step 4 Now we choose H so that dH[k+1] ω and dH[k+1] µ are O(k+2). This completes the proof of the ﬁrst item in Proposition 3. For the second item, we take the power solution above for an input map h and write

2 dH Θ = M(, h) = M0(h) + M1(h) + M2(h) + ... λ, 2 where Mi(h) are sections of the normal bundle of the image of h in H (X). We know that M0(h) is the mean curvature m(h), whose linearisation with respect to normal variations is a Jacobi operator Jh say. Locally, working over a ball,

16 we can ﬁnd a right inverse Gh to the Jacobi operator. Then starting with a solution h0 of the equation m(h0) = 0 we can construct a formal power series P k solution h = h0 + ξk to the equation M(, h ) = 0 in a standard fashion. For example the ﬁrst term is given by

2 h = h0 − Gh0 (M1(h0)) + O( ).

2.3 The Torelli Theorem for K3 surfaces The question of which compact 4-manifolds admit hypersymplectic structures is an interesting open problem. But for hyperkählerstructures there is a complete classification. The underlying manifold must be a K3 surface or a 4-torus and we restrict here to the former case. We recall the fundamental global Torelli Theorem, for which see [3], Chapter VIII for example. Proposition 4 Let X be the oriented 4-manifold underlying a K3 surface. 2 Write C for the set of classes δ ∈ H (X, Z) with δ.δ = −2 and write Diff0 for the group of diffeomorphisms of X which act trivially on H2(X). A hy- perkählerstructure on X determines a self-dual subspace H+ ⊂ H2(X, R) (the span of [ωi]) which has the two properties: • H+ is a maximal positive subspace for the cup product form. • There is no class δ ∈ C orthogonal to H+. Conversely, given a subspace H+ ⊂ H2(X; R) satisfying these two conditions there is a hyperkählerstructure with volume 1 realising H+ as the self-dual subspace and the structure is unique up to the action of Diff0 × SL(3, R). (In the last sentence, the hyperkählerstructure is regarded as an element of R3 ⊗ Ω2(X) and SL(3, R) acts on R3.)

Let h be a positive map from the 3-ball to H2(X; R). We say that h avoids -2 classes if for each point in B there is no δ in C orthogonal to the image of the derivative. This is an open condition on h. The definition extends in the obvious way to the sections of the flat bundle Hχ defined by a fibration X → M → B as considered above. Then a hyperkählerelement ω defines a positive section of Hχ which avoids −2 classes. A complete discussion of the converse brings in topological questions about the group Diff0 which we do not want to go into here. At least in the local situation, when B is a ball, we can say that a positive section which avoids −2 classes defines a hyperkählerelement.

3 Kovalev-Lefschetz ﬁbrations

Baraglia [2] shows that there are essentially no co-associative ﬁbrations over a compact base manifold B, so to get an interesting global problem we must allow singularities. The singularities that we consider here are those introduced by Kovalev [16], where the ﬁbres are allowed to develop ordinary double points.

17 Definition 1 A differentiable Kovalev Lefschetz (KL) fibration consists of data (M, B, L, L,˜ π) of the following form. • M is a smooth oriented compact 7-manifold, B is a smooth oriented compact 3-manifold and π : M → B is a smooth map.

• L ⊂ B is a 1-dimensional submanifold (i.e a link). • L˜ ⊂ M is a 1-dimensional submanifold and the restriction of π gives a diﬀeomorphism from L˜ to L. • π is a submersion outside L˜ and for any point p ∈ L˜ there are oriented charts around p, π(p) in which π is represented by a map

2 2 2 (z1, z2, z3, t) 7→ (z1 + z2 + z3 + E(z, t), t),

from R7 = C3 × R to R3 = C × R, where E(z, t) and its first and second partial derivatives vanish at z = 0. Thus the fibre π−1(p) of a KL fibration is a smooth 4-manifold if p is not in L, and if p is in L there is just one singular point, modelled on the quadric cone 2 2 2 3 2 {z1 + z2 + z3 = 0} in C , which is homeomorphic to the quotient C / ± 1. Now we define a torsion-free (respectively closed) co-associative KL fibration to be a differentiable one, as above, together with a smooth positive 3-form φ on M which is torsion-free (respectively closed) and such that the non-singular fibres are co-associative. We also require the orientations to be compatible with the form φ. A good reason for focusing on these kind of singularities is that Kovalev has outlined a programme that would produce many examples of torsion-free co-associative fibrations of this shape [16]. (Kovalev’s programme seems geometrically plausible, but there are some analytical difficulties in the available 2005 preprint [16]). One can certainly consider other kinds of singularities, but we will not do so here. Note that a compact manifold with holonomy the full group G2 has finite fundamental group [14]. Thus, up to finite coverings (and using the solution of the Poincaréconjecture), the essential case of interest is when the base B is the 3-sphere. We can now adapt our previous discussion (for genuine fibrations) to the KL situation. We will only consider the case when the smooth fibres are diffeomor- phic to a K3 surface. Let π : M → B be a differentiable KL fibration and γ be a small loop around one of the components of L. Then we can consider the monodromy around γ of the co-homology of the fibre. It is a well-known fact that this monodromy is the reflection in a “vanishing cycle’ δ which is a class in H2(X; Z) with δ.δ = −2. (In particular, this monodromy has order 2, so we do not need to discuss the orientation of γ.) To build in this monodromy we consider B as an orbifold, modelled on (R × C)/σ at points of L, where σ acts as −1 on C and +1 on R. Recall that an orbifold atlas is a covering of B by charts Uα ⊂ B such that if Uα intersects L

18 it is provided with an identification with a quotient U˜α/σ for a σ-invariant open set U˜α ⊂ R × C.A flat orbifold vector bundle over B with fibre a vector space V can be defined with reference to such an atlas as having local trivialisations of the form Uα × V over the charts which do not meet L and (U˜α × V, τα) over those which do, where τα is an involution of V . These trivialisations satisfy obvious compatibility conditions on the overlaps of charts. In our situation the cohomology of the fibres provides such a flat orbifold vector bundle H over B with the involutions τα given by the reflections in the vanishing cycles. We also want to consider flat orbifold affine bundles. These are defined in just the same way except that the τα are replaced by affine involutions of V Let V be any flat orbifold vector bundle over B. We can define a sheaf V of locally constant sections of V. In a chart Uα which intersect L such a section is, by definition, given by a τα-invariant element of V . Any flat orbifold affine bundle determines a flat orbifold vector bundle via the natural map from the affine group of Rn to GL(n, R). We will say that the affine bundle is a lift of the vector bundle. Extending the standard theory for flat bundles recalled in Section 2 above; if V is a flat orbifold vector bundle over B then there is a natural 1-1 correspondence between equivalence classes of flat orbifold affine bundles lifting V and elements of the cohomology group H1(B; V). Write Vχ for the affine bundle determined by a flat orbifold vector bundle 1 V and a class χ ∈ H (V). We define the sheaf of smooth sections of Vχ in the standard fashion; given over a chart (U˜α, τα) by smooth maps fα : U˜α → V satisfying the equivariance condition

fα(σ(x)) = τα(fα(x)). We will want to consider orthogonal vector and affine bundles. That is, we suppose that the model vector space V has a nondegenerate quadratic form of signature (p, q) and all the bundle data, and in particular each involution τα, is compatible with this. For our purposes we restrict to the case when p = 3. Away from L our bundle Vχ is an ordinary flat affine bundle so we are in the situation considered before and we have a notion of a positive section–i.e. the image of the derivative is a maximal positive subspace of the fibre. We want to extend this notion over L. To discuss this we will now fix attention on the case when each τα is a reflection in a negative vector. That is we assume that

τα(v) = v + (δα, v)δα + λαδα, where δα ∈ V with (δα, δα) = −2 and λα ∈ R. Let fα : U˜α → V be an equivariant map, where U˜α is some neighbourhood of the origin in R × C. Without real loss of generality, and to simplify notation, we can suppose that 2 λα = 0 and that fα(0) = 0. We write Dfα,D fα for the ﬁrst and second derivatives at 0.

Deﬁnition 2 We say that the fα is a branched positive map at (0, 0) if

1. Dfα vanishes on C, so the image of DFα is spanned by a single vector v0 ∈ V .

19 2 2 2 2. The restriction of D fα to C can be written as v1Rez + v2Imz where v1, v2 ∈ V .

⊥ 3. The vectors v0, v1, v2 span a maximal positive subspace in δ ⊂ V . In terms of local co-ordinates (w, t) on B, where w = z2, this can be expressed by saying that we have multi-valued function

3/2 2 (w, t) 7→ v0t + v1Rew + v2Imw + O(w , t ), with an ambiguity of sign in in the component along δ ∈ V . With this definition in place we can define a positive section h of our flat affine orbifold bundle Vχ to be a section which is positive away from L and represented by a branched positive map at points of L. We can define the 3-volume Vol3(h) of the section in the obvious way. Finally we can define a maximal positive section to be a positive section which is maximal in the previous sense away from L. These definitions imply that near points of L such a section yields a branched solution of the maximal submanifold equation, in the usual sense of the literature.

3.1 Proposed adiabatic limit problem Return now to a KL fibration π : M → B, the flat orbifold vector bundle H over B and the corresponding sheaf H. It is straightforward to check that H can be identified with the second derived direct image of the constant sheaf R on X and the first derived direct image is zero. So the Leray spectral sequence gives an exact sequence

0 → H3(B; R) → H3(M; R) →b H1(B; H) → 0.

Proposition 5 Suppose that φ0 is a closed positive 3-form on M with respect to which π : M → B is a co-associative KL ﬁbration and set χ = b([φ0]) ∈ 1 H ((B; H). Then φ0 determines a branched positive section of Hχ.

The proof is not hard but we will not go into it here.

We move on to the central point of this article. We say that a section h of a ﬂat aﬃne bundle Hχ avoids excess −2 classes if the following two conditions hold.

• The restriction to B \ L avoids −2 classes, in the sense deﬁned before. • At a point of L, and using the representation as in Deﬁnition 2 above, the only elements of C orthogonal to v0, v1, v2 are ±δα.

20 Write [B] ∈ H3(X) for the pull-back of the fundamental class of B. We makes a two-part conjecture.

3 Conjecture 1 Fix [φ0] ∈ H (M) and χ = b([φ0]).

1. If there is a positive section h of Hχ which avoids excess −2 classes, then for sufficiently large R there is a closed positive 3-form on M in the class [φ0] + R[B] with respect to which π : M → B has co-associative fibres. 2. If, in addition, the section h can be chosen to be maximal then, for sufficiently large R, we can choose the positive 3-form to define a torsion free G2-structure. These are precise conjectures which express part of a more general—but more vague—idea that for large R the whole 7-dimensional theory should be expressed in terms of positive sections of Hχ. For example we would hope that there be should be converse statements, going from the existence of the structures on M, for large R, to sections of Hχ. We can extend the discussion to Bryant’s Laplacian flow for closed positive 3-forms ∂φ = dd∗φ. ∂t The discussion in Section 2 suggests that the adiabatic limit should be the mean curvature flow for positive sections ∂h = m(h), (22) ∂t just as Hitchin’s volume functional relates to the volume functional Vol3h. Proving precise statements such as in Conjecture 1 will clearly involve substantial analysis. But we can leave that aside and formulate questions, within the framework of sections of flat orbifold bundles, which can be studied inde- pendently. Here we can take any orthogonal flat affine orbifold bundle Vχ with orbifold structure defined by reflections, as above.

1. Does Vχ admit positive sections? 2. If so, does it admit a maximal positive section?

3. If Vχ admits positive sections, is the volume functional Vol3 bounded above? 4. Does the mean curvature ﬂow (22) exist and converge to a maximal positive section?

Naturally we hope that these questions should have some relation with the corresponding questions for positive 3-forms at the beginning of this article. All of these questions should be viewed as tentative, provisional, formula- tions. We include two remarks in that direction.

21 • The condition of “avoiding excess −2 classes” seems very unnatural from the point of view of sections of the flat orbifold bundle Hχ. One could envisage that there could be cases where there are positive sections of Hχ which avoid excess -2 classes but where there is a maximal positive section which does not satisfy this condition. This might correspond to a torsion- free G2 structure on a singular space obtained by collapsing subsets of M. • In the definition of a Kovalev-Lefschetz fibration we required that there be at most one singular point in each fibre and the critical values form a union of disjoint circles. This should be the “generic” situation, but one could envisage that in generic 1-parameter families one should allow two critical points in a fibre. In that vein one could imagine that the long-time definition of the mean curvature flow (22) would require modifying the link L by allowing two components to cross, under suitable conditions on the monodromy of the flat bundle. Thus the right notion would not be a single KL fibration but an equivalence class generated by these operations.

4 More adiabatic analogues 4.1 Negative deﬁnite cup product

Let M be any compact 7-manifold with a torsion-free G2 structure deﬁned by a form φ. Suppose also that the holonomy is the full group G2, which implies that H1(M; R) = 0 [14]. Then it is known via Hodge Theory that the quadratic form on H2(M; R) deﬁned by

Q[φ](α) = hα ∪ α ∪ [φ], [M]i, is negative definite [14]. Now suppose that M has a C∞ Kovalev-Lefschetz fibration π : M → B with K3 fibres. The vanishing of H1(M) requires that 1 3 H (B; R) = 0. As in the previous section, let [φ0] be any fixed class in H (M) −1 and φR = [φ0] + R[B]. Then the limit of R Q[φR] as R tends to infinity is given by restricting to a smooth fibre X and taking the cup product on the fibre. Poincaréduality on B gives H2(B; R) = 0 and then the Leray spectral sequence shows that H2(M; R) = H0(H), which can be identified with the subspace of H2(X) preserved by the monodromy of the fibration. Now we have a related result in the context of positive sections.

Proposition 6 Let V be a flat orbifold bundle over a compact base B and Vχ be a lift to an affine orbifold bundle. Suppose that Vχ admits a positive section and that H1(B, R) = 0. If σ is a non-trivial global section of V then σ.σ < 0 at each point of B. We can write V = R σ ⊕ V0. The condition that H1(B; R) = 0 implies that χ is trivial in the R σ component, so if h is the positive section of Vχ then there is a well-defined continuous function σ.s on B which attains a maximum

22 at some point p ∈ B. Consider first the case when p is not in L. The maximality implies that σ(p) is orthogonal to the image of dh at p. By the definition of a positive section, this image is a maximal positive subspace which means that σ(p).σ(p) < 0. The definition of a positive section at points of L means that the same argument works when p ∈ L.

4.2 Associative submanifolds Recall that a 3-dimensional submanifold P ⊂ M in a 7-manifold with a positive 3-form φ is called associative if the restriction of φ to the orthogonal complement of TP in TM vanishes at each point of P . This implies that, with the right choice of orientation, the restriction of φ to TP is the volume form. These associative submanifolds are further examples of Harvey and Lawson’s calibrated submanifolds [10], with calibrating form phi and there is an elliptic deformation theory [17]. They are interesting geometrical objects to investigate in G2-manifolds: in particular there is a potential connection with moduli prob- lems, because if a non-zero homology class π ∈ H3(M) can be represented by a compact associative submanifold then we have h[φ], πi > 0. Let X be a K3 surface with a hyperkählertriple ω1, ω2, ω3. The cohomology + 2 classes [ωi] span a positive subspace H ⊂ H (X; R). Each non-zero vector τ = (τ1, τ2, τ3) defines a complex structure Iτ on X and a symplectic form P ⊥ + ωτ = τiωi. Write Hτ for the orthogonal complement of [ωτ ] in H . If Σ is a smooth complex curve in X, for the complex structure Iτ , then the product of Σ and the line Rτ is an associative submanifold in X × R3 with the positive P 2 3-form ωidti − dt1dt3dt3. In particular, suppose that c ∈ H (X; Z) is a class 2 ⊥ with c = −2 and c is orthogonal Hτ ; then standard theory tells us that there is a unique such curve Σ in the class c, and Σ is a 2-sphere. This leads to a natural candidate for an “adiabatic description” of certain associative submanifolds in a Kovalev-Lefschetz fibration. We begin with the local picture, away from the critical set L ⊂ B. Suppose that we have a positive section given by a map h into H2(X; R) and fix a class c as above. Let γ(s) be an embedded path in B so h◦γ is a path in H2(X; R) and for each s the derivative of h◦γ defines a complex structure on X and a 2-dimensional subspace H⊥(s) ⊂ Im dh ⊂ H2(X; R). If c is orthogonal to H⊥(s) for each s then we have a complex 2-sphere in the fibre over γ(s) and the union of these 2-spheres, as s varies, is our candidate for an adiabatic limit of associative submanifolds. The condition on the derivative of h ◦ γ has a simple geometric meaning. Up to parametrisation of γ, it is just the condition that γ be a gradient path for the function t 7→ c.h(t) with respect to the metric on B induced by h. To bring in the critical set L ⊂ B, recall that if p ∈ B \L is a point close to L there is a vanishing cycle Σ in the fibre Xp over p which is a 2-sphere bounding a “Lefschetz thimble” in M, and the homology class of the vanishing cycle is a class c as above. Let γ : [0, 1] → B be a path with γ(0) and γ(1) in L and γ(s) ∈ B \L for s ∈ (0, 1). For small > 0 there are vanishing cycles (defined up to isotopy) in the fibres over γ(), γ(1 − ) and we say that γ is a C∞ matching path if these agree under parallel transport along γ. In this case we can construct

23 an embedded 3-sphere in M by capping off the 2-sphere fibration over (, 1 − ) with the two Lefschetz thimbles. We say that γ is a homological matching path if the homology classes of the two vanishing cycles agree, a condition which is determined by the flat orbifold bundle H. For the present discussion we will not distinguish between the homological and C∞ conditions. Now we say that γ is a matching gradient path if it is a homological matching path and if over the interior (0, 1) it is a gradient path of the function h.c (interpreted using parallel transport of c along γ). These are plausible candidates for the adiabatic limits of certain associative spheres. More precisely, one can hope that, provided γ is a “nondegenerate” solution of the matching gradient path condition in a suitable sense, then there will be an associative 3-sphere in the G2 manifold for large values of R. On the other hand, the matching gradient paths are described entirely in terms of Hχ. The definition extends immediately to any flat orbifold bundle and they are objects which can be studied independent of any connection with G2 geometry. This proposed description of associative submanifolds is in the same vein as constructions of Lagrangian spheres in symplectic topology [1] and of tropical descriptions of holomorphic curves in Calabi-Yau manifolds [7].

5 Variants 5.1 Torus ﬁbres, special Lagrangian ﬁbrations

The local discussion of co-associative G2-fibrations in Section 2 applies with little change when the fibres are 4-tori rather than K3 surface. The “adiabatic limit data” is for a maximal submanifold in R3,3 = H2(T 4). The difference in this case is that such a maximal submanifold yields a genuine exact solutions of the equations for non-zero . This was the case treated by Baraglia in [2]. The source of this difference is that the curvature of the connection H vanishes in this situation—the horizontal subspaces are tangent to sections of the fibration. 1 Suppose now that M = S × N for a 6-manifold N, and consider G2- structures on M induced from SU(3) structures on N. Then we have φ = ρ + Ωdθ, where dθ is the standard 1-form on S1 and ρ, Ω are respectively a 3-form and a 2-form on N. Here we consider the case of torus fibrations M → B induced from special Lagrangian fibrations πN : N → B by taking a product of the fibres with a circle. Now H2(T 3 × S1) = H2(T 3) ⊕ H1(T 3) and the quadratic form on H2(T 3 ×S1) is induced from the dual pairing between H1(T 3),H2(T 3). In other words the reduction to the product situation yields a natural splitting ∗ R3,3 = R3 ⊕ R3 . The dual pairing also defines a standard symplectic form on R3,3 and the product structure implies that the image of the map h : B → R3,3 is Lagrangian.

24 It follows that this the image is the graph of the derivative of a function F on R3. The basic fact, observed by Hitchin [12], is that a Lagrangian submanifold of R3,3 defined as the graph of the derivative of a function F is a maximal submanifold if and only if F satisfies the Monge-Ampèreequation ∂2F det = C ∂xi∂xj for a constant C. This makes a bridge between the co-associative discussion and the extensive literature on special Lagrangian fibrations and “large complex structure” limits for Calabi-Yau 3-folds.

5.2 K3 fibrations of Calabi-Yau 3-folds Suppose again that M = S1 × N but now that B = S1 × S2 and consider 1 2 fibrations which are products with S of $N : N → S with K3 fibres. More precisely, we suppose that $N is a Lefschetz fibration in the usual sense, with a finite set of critical values in S2. The class [ω] gives a fixed class in the cohomology of the fibres so now we form a flat orbifold vector bundle H with fibre R2,19. The discussion of positive sections goes over immediately to this 2- dimensional case. We could hope that this has a bearing on the question: when can a differentiable Lefchetz fibration with K3 fibres be given the structure of a holomorphic fibration of a Calabi-Yau 3-fold? We can again make a bridge between this point of view and other, more standard, ones. Recall that the Grassmann manifold Gr+ of positive oriented 2-planes in R2,19 can be identified with an open subset of a quadric Q ⊂ CP20. This is achieved by mapping the plane spanned by the orthonormal frame e1, e2 20 2,19 to the complex line [e1 + ie2] ∈ CP . Let Σ ⊂ R be a surface with positive tangent space at each point. Then we have a Gauss map from Σ to Gr+. Just as in the Euclidean case, the maximal condition implies that this is holomorphic with respect to the induced complex structure on Σ; hence the image is a complex curve. Conversely, start with a complex curve in Gr+ ⊂ CP20, lift it locally to C21 and parametrise it by a complex variable z ∈ U ⊂ C. So we have holomorphic functions f0(z), . . . , f20(z) with Q(f0, . . . , f20) = 0. Let Hi be the dHi 2,19 integrals, solving dz = fi. Then the map from U to R with components 2,19 ReHi has image a maximal surface in R . This is the usual Weierstrasse construction. In our context, the complex curve in Gr+ amounts to a holomorphic description of a family of polarised K3 surfaces in terms of the Hodge data H2,0 ⊂ H2(X, C) and the Weierstrasse construction makes the bridge with the maximal submanifold equation.

5.3 Cayley fibrations Here we consider the local adiabatic problem for a fibration of an 8-dimensional manifold with holonomy Spin(7) by Cayley submanifolds. Recall that a torsion- free Spin(7) structure can be defined by a closed 4-form which is algebraically special at each point [14]. As the basic model we take two oriented 4-dimensional

25 Euclidean spaces V1,V2 and a ﬁxed isomorphism of the respective self-dual forms 2 ∗ Λ+(Vi ). Then we have a 4-form

X 0 Ω0 = dvol1 + ωa ∧ ωa + dvol2 on V1 ⊕ V2, where dvoli are the volume forms in the two factors, ωa runs 2 ∗ 0 over a standard orthonormal basis of Λ+(V1 ) and ωa the corresponding ba- 2 ∗ sis for Λ+(V2 ). A closed 4-form Ω on an 8-manifold M which is algebraically equivalent to Ω0 at each point defines a torsion free Spin(7) structure. A 4- dimensional submanifold X ⊂ M is a Cayley submanifold if at each point of X the triple (TMx, Ω(x),TXx) is algebraically equivalent to the model triple (V1 ⊕ V2, Ω0,V1). These are, again, examples of calibrated submanifolds [10] and there is an elliptic deformation theory [17]. A fibration π : M → B with Cayley fibres can be expressed in the manner of Section 2. There is a connection H which furnishes a decomposition of the forms on M into (p, q) components, and a closed 4-form Ω with three components

Ω = Ω4,0 + Ω2,2 + Ω0,4 The algebraic data is that at each point x of M we have oriented Euclidean structures on the horizontal and vertical subspaces, and an isomorphism between the spaces of self-dual forms, as above. The condition that Ω is closed can be viewed a set of coupled equations for the connection and the three components. A discussion parallel to that in the 7-dimensional case suggests that in the adiabatic limit the equations should de-couple, forcing the structure on the each fibre to be hyperkähler.This analysis leads us to formulate the following definition. Definition 3 Let H be a real vector space with a quadratic form of signature (3, q). Adiabatic Cayley data for H consists of an oriented Riemannian 4- manifold (B, g) and a closed H-valued self-dual 2-form Ψ on B such that Ψ∗Ψ 2 is the identity endomorphism of Λ+. To explain the notation here, at each point b ∈ B, the value Ψ(b) is regarded 2 2 ∗ as a linear map from the fibre of Λ+ to H (using the metric on Λ+). Then Ψ (b) is the adjoint, defined using the quadratic forms on each space, so the composite ∗ 2 Ψ Ψ is an endomorphism of Λ+. For our immediate application we take H to be the 2-dimensional cohomology of a 4-torus or a K3 surface X. Given adiabatic Cayley data as above, the image of Ψ(b) at each point b is a maximal positive subspace of H so defines a hyperkählerstructure on X. (More precisely this is defined up to diffeomorphism, but that is not important for the local discussion 2 here.) If ωi, ω2, ω3 is a standard orthonormal frame for Λ+(b) then we have a 0 −1 corresponding hyperkählertriple ωi of 2-forms on the fibre π (b). That is,

0 2 −1 [ωi] = Ψ(b)(ωi) ∈ H (π (b)). The sum X 0 Ω2,2 = ωa ∧ ωa

26 is independent of the choice of orthornormal frame and

df Ω2,2 = 0. Then is then a unique way to choose a connection H which is compatible with the volume form on the ﬁbres and so that

dH Ω2,2 = 0.

We take Ω0,4 to be extension of the volume form on the ﬁbres by this connection and Ω4,0 to be the lift of the volume form on the base. In this way we deﬁne a 4-form Ω on the total space of he correct algebraic type and with

dΩ = FH Ω0,4 + FH Ω2,2, terms which, as we argued before, should be suppressed in the adiabiatic limit. (When the fibre is a 4-torus then, just as in Baraglia’s case discussed above, the curvature FH is zero, so we can write down genuine torsion-free Spin(7)- structures, given adiabatic Cayley data. ) One remark about Definition 3 is that the metric g is entirely determined by the H-valued form Ψ. In other words suppose we are given an oriented 4- manifold B and an H valued 2-form Ψ on B. Then we can define Ψ∗Ψ, up to ambiguity by a positive scalar, using the wedge product form on 2-forms. Let us say that Ψ is special if at each point Ψ∗Ψ is a projection onto a 3-dimensional positive subspace in Λ2 (up to a factor). This subspace defines a conformal structure on B—the unique conformal structure so that this subspace is self- dual—and we fix a metric by by normalising so that Ψ∗Ψ is the identity on this subspace. In sum, an equivalent description of adiabatic Cayley data is a closed, special, H-valued 2-form.

5.4 Ricci curvature We recall ﬁrst a standard result. Proposition 7 Let V ⊂ Rp,q be a p-dimensional maximal positive submanifold. Then the Ricci curvature of the induced Riemannian metric on V is non-negative. We review the proof. By elementary submanifold theory, the Riemann curvature tensor of V is expressed in terms of the second fundamental form S. Fix a point in V and orthonormal frames for the tangent and normal bundles so S has a components Sij where i, j label the tangent frame and a the normal frame. Then the Riemann curvature tensor is X a a a a Rijkl = SikSjl − SjkSil, a where we have an opposite sign to the familiar Euclidean case because the quadratic form is negative in the normal direction. This gives

X a a X a a Rik = − (Sjj)Sik + SijSjk. a,j a,j

27 P a The term j Sjj is the mean curvature, which vanishes by assumption, and S is symmetric so we have X a a Rik = SijSkj, a,j which is non-negative. More precisely, the Ricci curvature in a tangent direction ξ to V is the square of the norm of the derivative of the Gauss map along ξ.

The relevance of this for us is an immediate consequence.

Corollary 1 Suppose that Hχ is a ﬂat aﬃne orbifold bundle over (B,L) which admits a maximal positive section. Then the induced Riemannian metric on B \ L has non-negative Ricci curvature.

We pass on to the higher dimensional case, where the result is less standard. Proposition 8 Suppose that (B, g, Ψ) is a set of adiabatic Cayley data. Then the Ricci curvature of g is non-negative.

2 We can regard Ψ as giving an isometric embedding of the bundle Λ+ over B into the trivial bundle with ﬁbre H. There is thus a “second fundamental form” S of this subbundle, or equivalently the derivative of the map from B to the Grassmannian. Let H⊥ be the complementary bundle in H. Then S is a tensor 2 1 ⊥ in Λ+ ⊗ Λ ⊗ H . A little thought shows that the condition that dΨ = 0 is equivalent to

2 • The connection on Λ+ induced from the Levi-Civita connection is the 2 same as the connection induced from the embedding Λ+ ⊂ H; 3 ⊥ 1 2 3 • The image of S in Λ ⊗ H induced the wedge product Λ ⊗ Λ+ → Λ is zero.

2 The ﬁrst item means that the Riemannian curvature of Λ+ can be computed from S. Indeed it is given by qH⊥ (S) where qH⊥ is the quadratic map from Λ+ ⊗ Λ1 ⊗ H⊥ to Λ2 ⊗ Λ+ by the tensor product of • The skew-symmetric wedge product Λ1 ⊗ Λ1 → Λ2;

2 2 2 • The skew-symmetric cross-product Λ+ ⊗ Λ+ → Λ+; • The symmetric inner product H⊥ ⊗ H⊥ → R. Of course to pin down signs it is crucial that the last form is negative definite, but otherwise it will not enter the picture. We define a quadratic map q from Λ+ ⊗ Λ1 to Λ2 ⊗ Λ+ by using the first two components above.

Next we want to recall how to compute the Ricci curvature of an oriented 2 Riemannian 4-manifold (B, g) from the curvature tensor F of Λ+. This curvature tensor F lies in (Λ+ ⊕ Λ−) ⊗ Λ+ so has two components, say

F+ ∈ Λ+ ⊗ Λ+ ,F− ∈ Λ− ⊗ Λ+.

28 We work at a fixed point b ∈ B and take a unit tangent vector e0. This choice of e0 induces an isomorphism between the self-dual and anti-self-dual forms at b. To fix signs, say that this isomorphism takes the self-dual form dx0dx1 +dx2dx3 to the anti-self-dual form −dx0dx1 + dx2dx3. Using this isomorphism we can define a trace

Tre0 :Λ− ⊗ Λ+ → R, depending on e0. Of course we also have an invariant trace:

Tr : Λ+ ⊗ Λ+ → R.

2 + We deﬁne Te0 :Λ ⊗ Λ → R to be the sum Tr + Tre0 on the two factors. The formula we need is that

Ric(e0) = Te0 (F ). This is easy to check, using the well-known decomposition of the curvature tensor of a 4-manifold, or otherwise. So in our situation

Ric(e0) = Te0 (qH⊥ (S)).

So we need to show that for any s ∈ Λ+ ⊗Λ1 which is in the kernel of the wedge + 1 3 product Λ ⊗ Λ → Λ we have Te0 (q(s)) ≤ 0. Calculate in an orthonormal frame ei extending e0. This induces a standard orthonormal frame ωi for Λ+. P3 P3 We can write s = i=1 vi ⊗ ωi, where vi = tie0 + i=1 sijej. One ﬁnds that the condition that s is in the kernel of the wedge product is that the the trace P sii is zero and that the ti are determined by the skew symmetric part of sij:

ti = sjk − skj for i, j, k cyclic. On the other hand one ﬁnds that for any s in Λ1 ⊗ Λ+ X Te0 (q(s)) = ti(sjk − skj). cyclic

P 2 So for s in the kernel of the wedge product this becomes − ti which has the desired sign In fact this calculation shows that again Ric(e0) is the square of the norm of the derivative of the map to the Grassmannian of 3-planes in H, in the direction e0.

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